Re: [math-fun] left vs. right
Forwarded from SBG ... ------- Date: Mon, 30 Aug 2010 12:08:08 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com> Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer. Steve Gray On 8/30/2010 11:44 AM, James Buddenhagen wrote:
Interesting. I always assumed that people who had trouble with maps, cardinal directions, etc. were not math people. Perhaps it is because I was around maps since a child that I have always puzzled that some people have trouble with them. When heavy fog or overcast I get lost much easier than on clear sunny days, but I do not consciously think about the sun or shadows.
Whereas visualizing geometric solids seems easy for me, this I assume is learned. Nonetheless, I am almost always confused by illustrations of how to insert cards in ATM machines, and have even resorted to first trying the opposite of what I think it says, my wife never has this problem. Also, she finds ambiguous directions that others give about how to get somewhere unambiguous, while I am weighing the odds of what interpretation to give. I guess people are just different.
Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com> Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer. Steve Gray ________________________________ It can't be done. Given one ray, v1, the other four must lie on the opposite side of the plane through P normal to v1. With v2 also chosen, the remaining three are confined to the inside of a dihedral with acute angle. With v3 also chosen, the remaining two are confined to the inside of a trihedral whose three dihedrals are acute. Let u1, u2, u3 be rays along the edges of the dihedral. Since v4 and v5 must be positive linear sums of u1, u2, u3, it suffices to prove that the dot products ui.uj are positive, for then v4.v5>0, and their angle cannot be acute. If we consider the spherical triangle intercepted by the trihedral, this becomes: "If the angles of a spherical triangle are acute, then their sides are acute." Note that the sides are measured by the angle they subtend at the center. I've pretty much forgotten my spherical trigonometry, so I will quote from ( http://mathworld.wolfram.com/SphericalTrigonometry.html ), eqs. (18)-(20). cos A = - cos B cos C + sin B sin C cos a, etc. by permutation. The angles A, B, C are acute, so their sines and cosines are positive. Whence cos a is positive, and so side a is acute, and similarly for the other sides, completing the proof of the assertion. -- Gene
On Mon, Aug 30, 2010 at 6:32 PM, <rcs@xmission.com> wrote:
Forwarded from SBG ...
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Date: Mon, 30 Aug 2010 12:08:08 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com>
Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer.
Not possible. Rotate the coordinates so that one ray passes through (0,0,1). Then the other 4 must all have negative z-components. Project these 4 onto the X-Y plane. The angle between two of these projections must be right or acute. The angle between the original rays whose projections form a nonobtuse angle will also form a nonobtuse angle. Andy
This problem appears as Problem 67 in "The Wohascum County Problem Book", Gilbert, Krusemeyer, Larson, MAA, 1993. ----- Original Message ----- From: <rcs@xmission.com> To: <math-fun@mailman.xmission.com> Cc: <rcs@xmission.com> Sent: Monday, August 30, 2010 5:32 PM Subject: Re: [math-fun] left vs. right
Forwarded from SBG ...
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Date: Mon, 30 Aug 2010 12:08:08 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com>
Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer.
Steve Gray
On 8/30/2010 11:44 AM, James Buddenhagen wrote:
Interesting. I always assumed that people who had trouble with maps, cardinal directions, etc. were not math people. Perhaps it is because I was around maps since a child that I have always puzzled that some people have trouble with them. When heavy fog or overcast I get lost much easier than on clear sunny days, but I do not consciously think about the sun or shadows.
Whereas visualizing geometric solids seems easy for me, this I assume is learned. Nonetheless, I am almost always confused by illustrations of how to insert cards in ATM machines, and have even resorted to first trying the opposite of what I think it says, my wife never has this problem. Also, she finds ambiguous directions that others give about how to get somewhere unambiguous, while I am weighing the odds of what interpretation to give. I guess people are just different.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
That's very curious. I thought the problem up myself as part of a larger inquiry, which I'm still working on. I'd buy the book but it costs $40 and I don't really have time for other problems. Steve Gray On 8/31/2010 7:22 AM, Loren and Liz Larson wrote:
This problem appears as Problem 67 in "The Wohascum County Problem Book", Gilbert, Krusemeyer, Larson, MAA, 1993.
----- Original Message ----- From: <rcs@xmission.com> To: <math-fun@mailman.xmission.com> Cc: <rcs@xmission.com> Sent: Monday, August 30, 2010 5:32 PM Subject: Re: [math-fun] left vs. right
Forwarded from SBG ...
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Date: Mon, 30 Aug 2010 12:08:08 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com>
Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer.
Steve Gray
On 8/30/2010 11:44 AM, James Buddenhagen wrote:
Interesting. I always assumed that people who had trouble with maps, cardinal directions, etc. were not math people. Perhaps it is because I was around maps since a child that I have always puzzled that some people have trouble with them. When heavy fog or overcast I get lost much easier than on clear sunny days, but I do not consciously think about the sun or shadows.
Whereas visualizing geometric solids seems easy for me, this I assume is learned. Nonetheless, I am almost always confused by illustrations of how to insert cards in ATM machines, and have even resorted to first trying the opposite of what I think it says, my wife never has this problem. Also, she finds ambiguous directions that others give about how to get somewhere unambiguous, while I am weighing the odds of what interpretation to give. I guess people are just different.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I hadn't meant to be accusatory -- obviously, the question is a natural one, and there is at least one Putnam problem which I can recall that is very similar. My point was simply to draw attention to the book, which is soon to come out is a second edition, which contains lots of very nice problems and solutions. Loren ----- Original Message ----- From: "Stephen B. Gray" <stevebg@roadrunner.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, August 31, 2010 5:03 PM Subject: Re: [math-fun] left vs. right
That's very curious. I thought the problem up myself as part of a larger inquiry, which I'm still working on. I'd buy the book but it costs $40 and I don't really have time for other problems.
Steve Gray
On 8/31/2010 7:22 AM, Loren and Liz Larson wrote:
This problem appears as Problem 67 in "The Wohascum County Problem Book", Gilbert, Krusemeyer, Larson, MAA, 1993.
----- Original Message ----- From: <rcs@xmission.com> To: <math-fun@mailman.xmission.com> Cc: <rcs@xmission.com> Sent: Monday, August 30, 2010 5:32 PM Subject: Re: [math-fun] left vs. right
Forwarded from SBG ...
-------
Date: Mon, 30 Aug 2010 12:08:08 -0700 Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com>
Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer.
Steve Gray
On 8/30/2010 11:44 AM, James Buddenhagen wrote:
Interesting. I always assumed that people who had trouble with maps, cardinal directions, etc. were not math people. Perhaps it is because I was around maps since a child that I have always puzzled that some people have trouble with them. When heavy fog or overcast I get lost much easier than on clear sunny days, but I do not consciously think about the sun or shadows.
Whereas visualizing geometric solids seems easy for me, this I assume is learned. Nonetheless, I am almost always confused by illustrations of how to insert cards in ATM machines, and have even resorted to first trying the opposite of what I think it says, my wife never has this problem. Also, she finds ambiguous directions that others give about how to get somewhere unambiguous, while I am weighing the odds of what interpretation to give. I guess people are just different.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (5)
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Andy Latto -
Eugene Salamin -
Loren and Liz Larson -
rcs＠xmission.com -
Stephen B. Gray